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Coulomb's Law is the essential formula used to calculate the electric force and electric field due to point charges. It states that the force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The mathematical expression is given by:
\[ F = \frac{k |q_1 q_2|}{r^2} \]where:

• \( F \) is the magnitude of the force between the charges,
• \( k \) is Coulomb's constant \( (8.99 \times 10^9 \text{ N m}^2/\text{C}^2) \),
• \( q_1 \) and \( q_2 \) are the magnitudes of the charges,
• \( r \) is the distance between the centers of the two charges.

When it comes to the electric field \( E \), produced by a charge \( Q \), Coulomb's Law adapts to:
\[ E = \frac{kQ}{r^2} \]Here, \( E \) represents the electric field at a point at a distance \( r \) from the charge. This field is the force per unit charge and acts along the line connecting the charge and the point of interest.

When a charge is uniformly distributed, it means that the charge is spread evenly over a surface or throughout a volume. In the context of a spherical volume, it suggests that the charge is distributed so evenly that any chosen section of the sphere holds the same charge density. The charge density \( \rho \) (charge per unit volume) for a volume charge is calculated by:
\[ \rho = \frac{Q}{V} \]where \( V \) is the volume of the sphere. For a sphere, this volume is given by \( V = \frac{4}{3} \pi R^3 \). Therefore, if a charge \( Q \) is uniformly distributed across a sphere of radius \( R \), the charge density becomes:
\[ \rho = \frac{Q}{\frac{4}{3} \pi R^3} \]Understanding this helps in deducing the electric field both inside and outside the sphere. Internally, within a charged sphere, the symmetry results in a net field of zero at any point inside, since contributions from all parts of the charge cancel out. Externally, the sphere behaves as if all its charge were concentrated at its center.

A spherical charge distribution refers to a setup where a charge is spread out evenly across the volume of a sphere. This creates certain symmetrical properties in how the electric field behaves both inside and outside the sphere.

• **Inside the Sphere:** The electric field inside a uniformly charged sphere is zero. This is because the charges are symmetrically placed around any given interior point, leading to uniform cancellation of forces from all directions.
• **Outside the Sphere:** For points beyond the surface of a charged sphere, the sphere behaves like a single point charge located at its center. This means the electric field at any external point at a distance \( r \) from the center is calculated using Coulomb's Law as if the sphere were merely a point charge:

\[ E = \frac{kQ}{r^2} \]This simplification is extremely beneficial when calculating fields along axes or in certain symmetric arrangements like in the given exercise.

The direction of an electric field is crucial in understanding how charges influence each other. By convention, the electric field direction is the direction that a positive test charge would move if placed in the field. For a positive charge, the electric field points radially outward, while for a negative charge, it points inward.
A simple understanding involves:

• **Positive Charges:** Generate an outward field away from the charge.
• **Negative Charges:** Produce an inward field towards the charge.
• **Influence of Multiple Charges:** When dealing with multiple charges, as in the exercise with two spheres, the total electric field at a point is the vector sum of the fields due to each charge distribution.

In the specific example of the spheres with charge \( Q \), each sphere contributes an electric field in a certain direction:

• At points on the x-axis, the field's direction is determined by comparing which sphere is closer or further, considering the signs and magnitudes of each contributing field.
• The net direction is the result of additive or subtractive interactions based on the field lines from each charge distribution.